Blue Peter presenter Helen Skelton decided on Thursday that the abuse had simply become too much too take. With a final sign-off to her 40,000 followers, she said goodbye to Twitter.

"Turns out I don't have very thick skin after all so I am closing my twitter account. Enjoy the games. Signing off, skelts x."

A day earlier, it had been the turn of Channel 4 presenter Kirstie Allsopp to air her frustration with so-called trolls posting abuse to her on Twitter. She said she had called in the police over vile and unacceptable messages from two anonymous Twitter users.

Allsopp refused to leave the social network over the comments, but told her 235,000 followers: "I will not accept being told to shoot my own womb or bleed to death with a spade in my vagina by anyone. Bullying is unacceptable."

The presenter, who fronts Location, Location, Location for Channel 4, said there should be a way to tackle internet trolls without calling in the police.

There appears to be a growing wave of abuse targeting high-profile personalities on Twitter, prompting heated debate about free expression on the internet.

Skelton, 29, did not retweet the comments that she had found insulting, but they are understood to be messages sent to her relating to her work on the Olympics, where she has been interviewing members of the public around the Games venues.

This week police in Dorset were criticised by libertarian groups when they arrested a 17-year-old on suspicion of malicious communications after threatening tweets to the British Olympic diver Tom Daley. The move was described as over the top, quick-triggered and motivated mainly by a public outcry in defence of the sportsman on Twitter.

Others have argued that the law must equally apply in cyberspace and that the internet is no different from a British high street. In May, a student who tweeted racist comments about the footballer Fabrice Muamba when he collapsed during a match was released after serving half a 56-day sentence imposed by magistrates in Swansea.

Internet trolls can face prosecution under legislation such as the Protection from Harassment Act 1997, Malicious Communications Act 1998 and Communications Act 2003.

Team GB's 18-year-old weightlifting star, Zoe Smith, "stuck two fingers up" to her internet critics on Monday when she lifted a British record-breaking 121kg. She spoke out about how she and teammates faced abuse for their appearance and choice of sport.

Smith said: "I love using Twitter. I am pretty much known as the athlete who sits there and pretty much tweets in between her clean and jerk. Some people just abuse it … Just because, at the moment perhaps while we are competing they think we are public figures, they think they can stay as faceless people who would say these things."

She was not alone. The swimmer Rebecca Adlington, who won bronze in the women's 400m freestyle final at the 2012 Games on Sunday, has been taunted on Twitter and footballers, including the England international Micah Richards and Everton's Darron Gibson, have ditched the social network after being abused.

The BBC's Gary Lineker said he felt physically sick reading some of the messages sent to his son, George, who suffered from leukaemia as a child. Lineker highlighted three tweets posted by an anonymous user. One said: "Heard ya Leukaemia, pity ya didn't die."

Other athletes have chosen to avoid the social network altogether. Louis Smith, Britain's bronze-winning gymnast, said: "It is just that one message. You can get 100 nice ones but there is that one message that you do not want to see – so that is why I took the decision to stop tweeting."

The abuse threatens to cast a shadow over Twitter, which has become known as the social network where big-name stars mingle with non-famous users. Stephen Fry, one of the site's earliest users, has repeatedly threatened to quit over abusive messages.

Twitter declined to comment about abusive behaviour on its social network. The company, which describes itself as the free speech wing of the free speech party, has said it does not monitor or intervene in messages posted by its 140 million active users. It encourages users to block people who they find insulting. The site's rules says users must not post direct, specific threats of violence against others.

But the company's hands-off approach contrasts with the recent interventions by police over Twitter messages.

Mark Williams-Thomas, a criminologist and child protection expert, accused Twitter of turning a blind eye to the darker corners of its site.

"To simply say we don't monitor content is irresponsible from a commercial operation and it's also morally completely wrong," he said.

Williams-Thomas said the law must apply online but it would put added strain on the already-stretched police force.

"My view is if you break the law and you are threatening and abusive to people on Twitter, you deserve to have the full weight of law come down on you," he said.

Stuart Hyde, the chief constable of Cumbria Constabulary and lead on e-crime for the Association of Chief Police Officers, said there police conducted inquiries into messages on Facebook on 14,000 occasions last year. That number will rise sharply if police are called in over comments on Twitter and other websites.

"People have a right to publish their views but when these views become indecent, threatening or offensive then the individuals they affect also have the right to report them," Hyde said.

"The public have a responsibility to keep their comments on social networks within the law. If they are not, then the police will assist with any prosecution."

• This article was amended on 3 August 2012 because the original said Darron Gibson was a Manchester United player.